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http://hdl.handle.net/10397/96256
Title: | Error analysis of a finite difference scheme for the epitaxial thin film model with slope selection with an improved convergence constant | Authors: | Qiao, Z Wang, C Wise, SM Zhang, Z |
Issue Date: | 2017 | Source: | International journal of numerical analysis and modeling, 2017, v. 14, no. 2, p. 283-305 | Abstract: | In this paper we present an improved error analysis for a finite difference scheme for solving the 1-D epitaxial thin film model with slope selection. The unique solvability and unconditional energy stability are assured by the convex nature of the splitting scheme. A uniform-in-time Hm bound of the numerical solution is acquired through Sobolev estimates at a discrete level. It is observed that a standard error estimate, based on the discrete Gronwall inequality, leads to a convergence constant of the form exp(CTε−m), where m is a positive integer, and ε is the corner rounding width, which is much smaller than the domain size. To improve this error estimate, we employ a spectrum estimate for the linearized operator associated with the 1-D slope selection (SS) gradient flow. With the help of the aforementioned linearized spectrum estimate, we are able to derive a convergence analysis for the finite difference scheme, in which the convergence constant depends on ε−1 only in a polynomial order, rather than exponential. | Keywords: | Epitaxial thin film growth Finite difference Convex splitting Uniform-in-time Hm stability Linearized spectrum estimate Discrete Gronwall inequality. |
Publisher: | Institute for Scientific Computing and Information | Journal: | International journal of numerical analysis and modeling | EISSN: | 1705-5105 | Rights: | © 2017 Institute for Scientific Computing and Information This is the accepted version of the following article: Qiao, Z., Wang, C., Wise, S. M., & Zhang, Z. (2017). Error analysis of a finite difference scheme for the epitaxial thin film model with slope selection with an improved convergence constant. International Journal of Numerical Analysis and Modeling, 14(2), 283-305, which has been published in https://www.global-sci.org/intro/article_detail/ijnam/421.html. |
Appears in Collections: | Journal/Magazine Article |
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